[Proj] +towgs84 approximation error

Noel Zinn (cc) ndzinn at comcast.net
Sat Mar 25 10:02:03 EST 2017

Hi Jochem,

Being new to this listserv you should know that there are tourists like 
myself who hang around for the geodetic tidbits and there are those who do 
the real work of maintaining proj4.  While I'm happy to support your 
project, it's the vote of the latter that counts.  I believe that you've 
made your case.

Having reread this thread, I agree there has been repetition.  Having said 
that, many excellent points were made early on.  Maybe they bear repeating.

Thanks for this view into your interesting work, Jochem!  " Does that 
explain why I say I need the strict formulas also for geodetic precision?". 
Well, yes it does if you are using the strict formulas for the derivation of 
your transformation.  Must make for messy linearized observation equations. 
But 40 arc seconds of rotation is large.  I presume this isn't a country 
worked by C.F. Gauss!

The high parametric correlation that we agree upon has a practical 
consequence.  I presume that you will publish your datum shift and the 
translations will be rounded to the nearest millimeter, which is the 
numerical accuracy you mentioned earlier, and not to the full precision left 
over in your computer after the derivation.  From previous studies of 
small-area derivations I know that the dilution of precision (DOP) for the 
translations will be at least 10 and as much as 100 (or more for a country 
the size of Cyprus).  So, I expect that an average publication truncation 
error of 0.0005m will escalate to 0.005 or 0.05 (or more) at the surface 
depending upon where you're working.  Something to consider with your tight 
error budget, but I'm sure that you have.  BTW, the translation DOP for a 
3-parameter shift must be about 1.

Regarding your revised challenge, I was hoping for somewhat more geodetic 
(stochastic) inaccuracy than 1cm to better absorb numerical differences 
between the small-angle approximation and your fully-populated rotation 
matrix.  But your suggestion is interesting nonetheless.  I would indeed 
like to know how well regression equations can model a 40 arc second 
rotation (with minimal parameters).  So, I'm up for it.

All the best,

Noel Zinn, Principal, Hydrometronics LLC
+1-832-539-1472 (office), +1-281-221-0051 (cell)
noel.zinn at hydrometronics.com (email)
http://www.hydrometronics.com (website)

-----Original Message----- 
From: Jochem
Sent: Saturday, March 25, 2017 8:18 AM
To: proj at lists.maptools.org
Subject: Re: [Proj] +towgs84 approximation error

Hi Noel,

You mean you would like to see less rapid, but more cheery responses? I am
Sorry. I really do find the discussion with you and the others very
interesting en enjoyable. But, I notice we both tend to repeat ourselves,
which is not getting us somewhere. I'll try to break with that.

I am happy you support my suggestion of having the strict formulas as an
option. You are totally right that it would not be more accurate in a
geodetic sense, in most cases. However, are cases where the strict formulas
are more accurate, also geodetically. I'll explain.

The country where I was working this week has an imprecise (20 cm errors)
old triangulation network with large (40 arc-seconds) rotation parameters to
ITRF2008 at 2005.0=WGS84(G1762). They installed a continuously operating GNSS
reference station (CORS). Using that, they corrected the coordinates of
their first order triangulation points and changed the false Northing and
Easting to not mix corrected and uncorrected coordinates. By that they
introduced a new precise reference frame, but with the same large rotation
parameters to WGS84. The precise values of these parameters are still
unknown. But they started to log the GNSS observations of their CORS. As
soon as they acquired a full year of data, I will compute the precise
coordinates of their CORS relative to the IGS network and estimate the
transformation parameters to ITRF2014 using adjustment software that happens
to use the strict formulas. And I will continue to do that yearly. The
precision of this transformation to ITRS will be definitely be better than
the 4 cm approximation error of the default +towgs84 formulas. Does that
explain why I say I need the strict formulas also for geodetic precision?

As I said I agree totally with your point that the 7 parameters are often
highly correlated, but I am doubting if that influences the precision of
transformed coordinates. All the other things you say are correct, no
doubts. An experiment (or challenge if you wish) would be very interesting.

However, the challenge you suggest makes no sense to me. We both know that
old triangulation frames have a too low accuracy to make the strict formulas
necessary  (unless it would have enormous > 1 arc-minute rotations). So I
would like to alter the experiment on two points: use a GNSS corrected old
frame (like pseudo-RD of the Netherlands) with 1 cm relative standard
deviations, but with rotation angles to ETRS89 up to 40 arc-seconds (I can
just shift the Netherlands 1 km to the north to accomplish that, to simulate
an error of 1 km in the astronomical observations of the central point)...

Kind regards, Jochem

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